Principal functions of impulsive difference operators on semi axis

In this paper, we investigate the continuous spectrum and resolvent operator of a second-order difference operator with an impulsive condition. Then, under certain conditions, we prove finiteness of eigenvalues, spectral singularities. At last, we present principal functions of corresponding impulsive operator.

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  • Samoilenko, A. M. and Perestyuk, N. A., Impulsive differential equations, World Scientific, Singapore, 1995.
  • Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations, World Scientific, Singapore, 1998.
  • Bainov, D. D. and Simeonov, P. S., Oscillation theory of impulsive differential equations, Int. Publ., Orlando, 1998.
  • Uğurlu, E. and Bairamov, E., Spectral analysis of eigenparameter dependent boundary value transmission problems, J. Math. Anal. Appl., 413, 1, (2014), 482--494.
  • Mostafazadeh, A., Spectral singularities of a general point interaction, J. Phys. A. Math. Theory, 44, 375302, (2011), (9pp).
  • He, Z. M. and Zhang, X. M. , Monoton iterative technique for first order impulsive difference equations with periodic boundary conditions, Appl. Math. Comput., 156, 3, (2004), 605--620.
  • Wang, P. and Wang, W., Boundary value problems for first order impulsive difference equations, Int. Journal of Difference Equations, 1, (2006), 249--259.
  • Zhang, Q., On a linear delay difference equations with impulses., Annals of Differential Equations, 18, 2, (2002), 197--204.
  • Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operators of second order on a semi-axis, AMS Transl. (2), 16, (1960), 103-193.
  • Marchenko, V. A., Sturm-Liouville operators and applications, Birkhauser Verlag, Basel, 1986.
  • Levitan B. M. and Sargsjan I. S., Sturm-Liouville and Dirac operators, Kluwer Academic Publishers Group, Dordrecht, 1991.
  • Agarwal, R. P., Difference equations and inequalities, in: Theory, Methods and Applications, Marcel Dekkar Inc., New York, Basel, 2000.
  • Kelley, W. G and Peterson, A. C., Difference equations: an introduction with applications, Harcourt Academic Press, 2001.
  • Krall, A.M., Bairamov, E. and Cakar, O., Spectral analysis of a non-selfadjoint discrete Schrödinger operators with spectral singularities, Math. Nachr., 231, (2001), 89--104.
  • Bairamov, E., Cakar, O. and Krall, A.M., Non-Selfadjoint Difference Operators and Jacobi Matrices with Spectral Singularities, Math. Nachr., 229, (2001), 5--14.
  • Adıvar, M. and Bairamov, E., Spectral Properties of Non-Selfadjoint Difference Operators, J. Math. Anal. and Appl., 261, (2001), 461--478.
  • Adıvar, M. and Bairamov, E., Difference Equations of Second Order with Spectral Singularities, J. Math. Anal. Appl., 277, (2003), 714--721.
  • Guseinov, G. Sh., The inverse problem of scattering theory for a second order difference equation, Sov. Math., Dokl., 230, (1976), 1045-1048.
  • Olgun, M., Köprübaşı, T. and Aygar, Y., Principal functions of non-selfadjoint difference operator with spectral parameter in boundary conditions, Abstr. Appl. Anal., Art. ID 608329, 10, (2011).
  • Erdal, I., Yardımcı, S., Eigenvalues and Scattering Properties of Difference Operators with Impulsive Condition, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68, 1, (2019), 663--671.
  • Lusternik, L.A, Sobolev V. I., Elements of Functional Analysis, Halsted Press, New York, 1974.
  • Glazman I. M., Direct Methods of Qualitative Spectral Anaysis of Singular Differential Operators, Jerusalem, 1965.
  • Dolzhenko E. P., Boundary value uniqueness theorems for analytic functions, Math. Notes, 26, (1979), 437-442.